Numerical analysis of penetration lengths in submerged supercritical water jets

J. of Supercritical Fluids 82 (2013) 213–220

Contents lists available at ScienceDirect

The Journal of Supercritical Fluids journal homepage: www.elsevier.com/locate/supflu

Numerical analysis of penetration lengths in submerged supercritical water jets Martin J. Schuler, Tobias Rothenfluh, Philipp Rudolf von Rohr ∗ ETH Zurich, Institute of Process Engineering, Sonneggstrasse 3, CH-8092 Zurich, Switzerland

a r t i c l e

i n f o

Article history: Received 5 May 2013 Received in revised form 30 July 2013 Accepted 31 July 2013 Keywords: Supercritical water jet Hydrothermal spallation drilling Penetration length Numerical model Entrainment Heat transfer

a b s t r a c t “Hydrothermal spallation drilling” is a possible alternative drilling technology that uses the properties of certain rock types to disintegrate into small fragments when heated up rapidly by a hot impinging fluid jet. Hot supercritical water jets are favored to provide the required heat for thermal rock fragmentation. However, the indispensable presence of a dense water-based drilling fluid during operation can cause considerable heat losses in the supercritical water jet before impingement on the rock surface. To predict these heat losses from the hot jet to the cold aqueous environment, a numerical model based on the commercial CFD tool ANSYS FLUENT® was established. Penetration lengths of the supercritical jet plume at near-critical pressures were determined numerically and validated with experimental values for a wide range of conditions. Experiments and simulations showed an acceptable agreement and the experimental trends were satisfactorily predicted by the model. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Various experimental and numerical investigations of submerged water jets over a wide range of temperature (T) and pressure (p) conditions are available in literature, as water is the most important component in industry and nature. Water jets can be divided into liquid jets, steam jets and supercritical (p > 220.64 bar, T > 373.95 ◦ C) jets [1]. To the authors’ knowledge, only two experimental investigations of round supercritical water jets injected into a cold co-flowing water environment are available in literature. In the experiments reported by Augustine [2], axial temperature profiles along the central axis of the jet were measured to investigate the thermal field. The focus in this study was on the jet’s far-field. In the comprehensive experimental study of Rothenfluh et al. [3], supercritical water (SCW) jets surrounded by a slowly co-flowing, subcritical (p > 220.64 bar, T < 373.95 ◦ C) cooling water (CW) stream (20 ◦ C) were discharged from a round nozzle in gravitational direction. The experiments were conducted inside a high pressure vessel at an absolute pressure of 224 bar. This setup was used to investigate the heat transfer between the hot, supercritical water jet and

∗ Corresponding author. Tel.: +41 44 632 92 63; fax: +41 44 632 13 25. E-mail addresses: [email protected] (M.J. Schuler), rothenfl[email protected] (T. Rothenfluh), [email protected], [email protected] (P. Rudolf von Rohr). URL: http://www.ltr.ethz.ch (P. Rudolf von Rohr). 0896-8446/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.supflu.2013.07.017

the surrounding cooling water. The axial extent of the supercritical water jet from the nozzle orifice to the axial position where the jet has cooled down to the pseudo-critical temperature (PCT) of water is called penetration length (PL). In reference [3], these penetration lengths were measured for a wide range of experimental conditions and different nozzle diameters to quantify entrainment and turbulent mixing effects between the high-velocity hot jet and the nearly stagnant cold environment. The experiments showed that penetration lengths are almost equal to the injector’s nozzle diameters and virtually independent of the jet’s nozzle exit temperature and the jet’s supercritical water mass flow rates for almost all experimental conditions applied. For an operating pressure of 224 bar, the PCT of water is found at 375.21 ◦ C and is therefore close to the critical one (373.95 ◦ C) [1]. Inside the hot supercritical plume (T > PCT), water is in a gas-like state, whereas the cold surrounding cooling water (T < PCT) is in a liquid-like state [1,3]. Hence, in the vicinity of the supercritical jet’s “boundary”, the temperature of the water is close to the PCT and the water’s thermo-physical properties show a strong local variation. The temperature and pressure level (375.21 ◦ C, 224 bar), where these significant changes of the thermophysical properties of water occur is also known as pseudo-critical point (PCP). This also includes sharp density variations with consequent strong changes in the refractive index [1] that were used by Rothenfluh et al. [3] to optically determine the penetration lengths of the supercritical jets. A detailed description of the experimental setup and the applied methods can be found in reference [3]. Due to the limited experimental possibilities of characterizing high-temperature and high-pressure supercritical water flows,

214

M.J. Schuler et al. / J. of Supercritical Fluids 82 (2013) 213–220

computational fluid dynamics (CFD) are widely used as a powerful tool for gaining deeper insight into such flows [4–7]. Whenever turbulence models originally developed for constant property flows are to be used for significantly variable property flows at temperatures close to the PCP, the applicability of these models is critically discussed in literature and results should be viewed with care [8–10]. In the work of Sierra-Pallares et al. [11], the authors made an attempt to numerically model the mixing of submerged supercritical water jets. The Peng–Robinson equation of state with volume translation and the NIST (National Institute of Standards and Technology) database were used to calculate the fluid properties. Moreover, the influence of intrusive, thermocouple based temperature measurements are addressed. Numerically determined axial temperature profiles were finally validated with the experimental data of Augustine [2]. Further challenges in simulating SCW systems at near-critical pressures with strongly varying thermophysical properties (REFPROP® database) are discussed in a former work of the article’s authors [12]. The focus of this work was on the numerical modeling of the axial temperature profile of submerged, supercritical water jets discharged in a subcritical environment of cooling water. Experimental axial temperature measurements were used to validate the applicability of common turbulence models to the simulation of these jets. The state-of-the-art turbulence models, however, have been developed for flow with constant or moderately varying fluid properties and assume a constant value for the turbulent Prandtl number (Prt ). Using these models for the calculation of the turbulent heat transfer around the PCP led to deviations between simulation and experiment, especially in the thermal far-field of the jet. Different concepts of implementing a locally varying turbulent Prandtl number within the flow field were developed to overcome the discrepancies between simulation and experiment of submerged supercritical water jets in the thermal far-field [12]. In contrast to the two theoretical publications mentioned above, the present work shows numerically determined values of the supercritical penetration length (PL) in comparison to non-intrusive optical measurements. Moreover, this study covers a wide range of geometrical (different nozzle diameters) and operating conditions (different nozzle exit temperatures and SCW mass flow rates). The possibility of applying hot water jets for the promising, alternative drilling technology called “hydrothermal spallation drilling” (HSD) [2] is the motivation behind this fundamental study. For drilling wells of several kilometers depth, the use of a (water-based) drilling fluid is indispensable and fulfills numerous important tasks in the drilling process (e.g. removal of rock cuttings). In a water-filled borehole, the hydrostatic pressure exceeds the water’s critical pressure (220.64 bar) beyond a certain depth. Hence, a promising approach could be to apply submerged, supercritical water jets as heat source for HSD [3,13]. In the drilling process, a hot fluid jet impinges on the upper rock layer at the bottom of the borehole. Due to the low thermal conductivity of the impinged rock, the upper rock layer can be rapidly brought to high temperatures. Thermal expansion of this hot layer cause high compressive stresses that finally leads to material failure and the formation of small fragments (spalls). The continuous removal of cuttings from the rock bulk by the heating process results in drilling progress [14–18]. However, heat losses of the supercritical water jet toward the aqueous and cold drilling fluid can negatively affect the efficiency of the spallation process, since a part of thermal energy is lost to the drilling fluid before it can be transferred to the rock. It is mainly the so-called entrainment of the cold, surrounding fluid into the hot jet region that is responsible for these heat losses [3]. In the present work, the commercial CFD tool ANSYS FLUENT® (12.1.4) was used to model these heat transfer phenomena of supercritical water jets submerged in a cold aqueous environment. The Favre-averaged conservation equations of momentum,

mass and energy were the basis of the model [19,20]. For round jets at high Reynolds numbers, the realizable k–␧ (rke) turbulence model is suggested [21]. All thermo-physical properties of water were implemented in the model by user-defined functions (UDF) based on the REFPROP® 8.0 database [22]. Finally, all numerical results were validated with the optical measurements performed by Rothenfluh et al. [3]. 2. Numerical model 2.1. Conservation equations and turbulence model The conservation equations for momentum and mass were solved as Favre-averaged Navier–Stokes equations [19,20] closed with the realizable k–ε turbulence model [21]. In this model, two additional transport equations for the turbulent kinetic energy (k) and its dissipation rate (ε) have to be solved. For round turbulent jets at high Reynolds numbers as the one under investigation, the realizable k–ε turbulence model [21] is recommended in literature, because it resolves the “round-jet anomaly” [23]. Unlike other common eddy viscosity turbulence models, the realizable k–ε model is able to predict not only the spreading rates of plane jets reasonably well but also for round axisymmetric jets. This is mainly due to a novel formulation of the dissipation rate equation (ε) [21,23]. Also for the description of flows in the supercritical state, the realizable k–ε turbulence is a common choice [11,24,25]. Finally, also the energy conservation equation was needed to describe heat transfer phenomena of hot water jets. For almost all constants appearing in the used realizable k–ε turbulence model, the default values suggested by ANSYS FLUENT® were applied. A value of 0.7, however, was assigned to the turbulent Prandtl number (instead of the default value 0.85), since values for Prt between 0.6 and 0.7 are recommended for fully developed hot round jets [26]. 2.2. Computational domain It is mainly the hot, developing region of the fluid jet close to the nozzle orifice that is of major interest for HSD, because rock surface temperatures above 350 ◦ C are required to achieve spallation [15,27,28]. The axisymmetric simulation domain shown in Fig. 2 is divided into solid parts (injection system) and fluid parts (water flow). The domain was chosen large enough to accurately simulate both the developing jet region, but also its far-field. All solid parts of the jet injection system were made of stainless steel 1.4435 (316 L). An air gap insulation between the central hot water stream and the cold annular water flow minimized heat losses from the hot to the cold water (see Fig. 2). Injection systems with nozzle diameters of 2 mm, 3 mm and 4 mm were used. Apart from explaining the computational domain for a diameter d0 of 3 mm, Fig. 2 also features a typical temperature contour plot of the jet flow considered. In all simulations, structured meshes were applied. A principle scheme of the used meshes is supplementary illustrated in Fig. 2 by magnifications of the underlying mesh defined as (a)–(e). The mesh in the nozzle (see magnification (b) in Fig. 2) itself and in the near-field of the jet (see dotted area in Fig. 2) has a basic resolution of 32 cells/mm (quadratic cells). Toward the inlet and outlet in axial direction and toward the confining outer wall in radial direction, the cells are smoothly growing in size that finally resulted in rectangular cells (magnification (a) and (c) in Fig. 2) and enlarged quadratic cells (magnification (d) and (e) in Fig. 2). The total number of cells was higher for small nozzle diameters to capture the higher gradients of the velocity and temperature field. The algorithm applied to guarantee a mesh-independent solution is described in Section 2.4. For a nozzle diameter of 2 mm, 3 mm and 4 mm, respectively, meshes with ∼1,365,000, ∼875,000 and

M.J. Schuler et al. / J. of Supercritical Fluids 82 (2013) 213–220

Supercritical water (SCW)

.

mCW

T

CW

8

8

8

u CW

.

T0 m0 h0 ρ0 u0 G 0

8

CW

8

8

A

ρCW

Cooling water (CW) CW G 8

CW

h

d0 A0

Nozzle exit

Injector

PL

Entrainment

Supercritical plume T = PCT r Spreading Gravity

x

Fig. 1. Simplified drawing of a round submerged supercritical water jet including the definitions of main parameters for the penetration length study. The temperature of the hot water jet at the nozzle exit and the temperature of the co-flowing CW ˙ , u, h and G are the mass , respectively. m, subcritical cooling water are T0 and T∞ flow rate, density, velocity, specific enthalpy and momentum flow at the corresponding temperature and pressure. d0 is the diameter of the nozzle orifice and A0 represents the corresponding area of the outlet nozzle. ACW ∞ is the cross-sectional area of the annular cooling water outlet. The distance from the nozzle outlet to the position of the pseudo-critical temperature (PCT) on the jet’s axis is defined as penetration length (PL).

∼615,000 cells were finally used. Heat transfer in the solid material was considered in the whole domain. The Enhanced Wall Treatment of ANSYS FLUENT® including the required mesh refinement criteria in wall normal direction were applied. Mass-flow inlet boundary conditions for all incoming water streams into the reactor were chosen. The option pressure-outlet was set as boundary condition for the effluent flow. A detailed description of the experimental setup is provided by Rothenfluh et al. [3]. More information about the used computational domains can be found in reference [12]. 2.3. Material properties The thermo-physical properties of water were implemented into the numerical model over the required temperature range for a pressure of 224 bar (operating pressure). The application of isobaric thermo-physical properties is justified, because local pressure variations in the flow had a marginal influence on the properties of water and finally on the simulated results [12]. Tabulated

215

property data versus temperature with a resolution of 0.1 K and linear interpolation in between were used in the numerical model. The evaluation of the water’s properties (density (), thermal conductivity (), dynamic viscosity (), isobaric heat capacity (cp ) and speed of sound (a)) was based on the REFPROP® 8.0 database [22]. The uncertainties of these property values for the applied temperature and pressure range can be found in [1,22]. For the pressure of 224 bar applied in the experiments, the PCT of water is 375.21 ◦ C. All thermo-physical properties of water undergo strong and sharp variations in the vicinity of the PCT for near-critical pressures. Crossing the PCP, water changes from a liquid-like state into a gas-like state with increasing temperatures. Hence, also the thermo-physical properties change from a liquid-like to a gas-like behavior [1,12,22]. Constant material properties according to the datasheet of the supplier were assigned to the solid parts (steel, 1.4435) of the simulation domain depicted in Fig. 2. Properties of air in the air gap isolation of the injection system (see Fig. 2) were calculated as a function of temperature at atmospheric pressure [22]. All material properties were finally implemented in the CFD code by user defined functions (UDF).

2.4. Solver settings and solution procedure In ANSYS FLUENT® , the Pressure-Based Coupled Solver was chosen for all simulations. Viscous heating and gravitation (9.81 m/s2 ) were considered in the model. Steady-state simulations with second-order schemes for spatial discretization were performed. The Least Squares Cell Based method was applied for gradient evaluation. All simulations were parallelized on the Brutus Cluster of ETH Zurich. Due to the strongly varying thermo-physical properties of water, numerical difficulties around the pseudo-critical point are always an issue. The initial solution is calculated with the standard k–ε turbulence model including constant fluid properties. Afterwards, all routines to determine temperature dependent properties are consecutively applied till a final converged solution is reached. This solution is used to start the simulation with the chosen realizable k–␧ turbulence model. Generally, the default under-relaxation factors recommended by ANSYS FLUENT® had to be reduced in order to reach a convergent solution: A reduction in the range of 25–75% compared to the default values was required to guarantee numerical stability and thus a convergent iterative solution. The axial distance between the nozzle orifice and the position of the PCT (375.21 ◦ C at 224 bar) on the jet’s axis is defined as penetration length (PL, see Fig. 1) in the context of this article. This PL defines the length of the supercritical part of the jet, where water is in a gas-like state (T ≥ 375.21 ◦ C).

Fig. 2. Temperature contour plot in the 2D-axisymmetric computational domain for a nozzle diameter of 3 mm including all required features. Mass flow rates of 4 g/s for SCW and 65 g/s for CW were applied at a pressure of 224 bar. A principle scheme of the used structured mesh is also given by magnifications of the underlying mesh named (a)–(e), which show the general shape of the cells. The dotted area highlights the region where additional mesh refinement was especially needed (Dynamic Gradient Adaption Approach).

M.J. Schuler et al. / J. of Supercritical Fluids 82 (2013) 213–220

Penetration length (PL) [mm]

216

Table 1 Comparison between a PL value determined experimentally and PL values obtained in the CFD simulations using the available turbulence models of ANSYS FLUENT® (12.1.4). All values were obtained for a nozzle diameter of 3 mm and an operating pressure of 224 bar. Mass flow rates of 4 g/s (SCW) and 65 g/s (CW) were used. The cooling water and nozzle exit temperature was set to 20 ◦ C and 452 ◦ C, respectively. The nozzle exit Reynolds number was approximately 60,000.

6 Mesh study for d 0 = 3mm

5 4

PL [mm]

3 2 1 0 0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Experiment (optically determined)

2.77

Turbulence model of ANSYS FLUENT® Standard k–ε model (ske) Renormalization-group (RNG) k–ε model Realizable k–ε model (rke) Standard k–ω model (skw) Shear-stress transport (SST) k–ω model Reynolds stress model (RSM) (Stress-Omega)

5.25 7.42 2.84 6.78 5.26 4.42

6

Number of computational cells *10

Fig. 3. Penetration length (PL) plotted as a function of the number of cells within the computational domains. Each data point represents the converged solution (given by PL) of a single CFD simulation with an individual mesh. The nozzle diameter for all simulations was 3 mm. The Dynamic Gradient Adaption Approach of ANSYS FLUENT® was used for mesh refinement. The realizable k–ε turbulence model was chosen. Mass flow rates of 4 g/s (SCW) and 65 g/s (CW) at an operating pressure of 224 bar were applied. The cooling water and nozzle exit temperatures were set to 20 ◦ C and 475 ◦ C, respectively.

Extended mesh convergence studies were performed for all simulated nozzle diameters (2 mm, 3 mm, 4 mm). For these studies, a conservative approach was chosen by simulating the experimental conditions that yielded the largest gradients within the velocity and temperature field. Finally, a mesh-independent solution was achieved for all nozzle diameters by applying solution-adaptive mesh refinement in ANSYS FLUENT® (Dynamic Gradient Adaption Approach). Mesh refinement was applied in regions of the computational domain, where the normalized local gradients of selected variables (e.g. k) exceeded a pre-defined threshold. The introduction of additional cells within these regions was controlled and limited by a pre-set value for the maximum allowed level of refinement. The performed mesh convergence studies showed that the computational mesh had to be vigorously refined in the shear layer of the jet after the nozzle orifice (see Fig. 2, dotted area). Especially cells with temperatures close to the pseudo-critical point appearing within this shear layer had to be significantly refined to account for the strongly varying fluid properties. Selected variables and parameters characterizing the flow (e.g. potential core length, PL, . . .) were recorded in the simulations to guarantee a iteration-independent and mesh-independent solution. The procedure mentioned above was time consuming and computationally expensive. Fig. 3 shows the results of the mesh convergence studies for a nozzle diameter of 3 mm: Each data point represents the converged solution given by PL of a single CFD simulation. To compare the results of each run, the length of the supercritical jet (PL) was evaluated for each simulation and plotted against the number of cells used in the computational domain. The simulations were performed on different meshes with different numbers of cells obtained by the refinement algorithm explained above. Fig. 3 shows that the PL levels off for a sufficiently high mesh refinement and hence becomes mesh-independent. Thus, for the nozzle diameter of 3 mm, a mesh containing a total of ∼875,000 cells was finally used for all simulations. 3. Results and discussion Penetration lengths of supercritical, submerged water jets were determined optically in the comprehensive experimental study of Rothenfluh et al. [3]. The boundary conditions for the jet flow applied in the simulations of the present work exactly correspond

to the experimental conditions used in the measurements according to reference [3]. Due to heat conduction in the solid parts of the injector, heat losses occur in the SCW stream on the way to the nozzle exit. The thermal inlet boundary condition of the SCW flow (see Fig. 2, Inlet SCW) had to be adjusted iteratively in the simulations to finally match the desired (iteration-independent) temperature at the nozzle exit (T0 ) measured in the experiment. The position of the PCT on the axis of the jet was extracted from the simulation data to determine the penetration length (PL). In Table 1, one value for the PL measured in [3] is compared to different state-of-the-art turbulence models of ANSYS FLUENT® for a nozzle diameter of 3 mm. For the constants in the used turbulence models, the default values suggested by ANSYS FLUENT® were applied. Only for the turbulent Prandtl number (Prt ), however a value of 0.7 deviating from the default value (0.85) was used [26]. Table 1 shows that all applied turbulence closure approaches overestimate the value of the PL obtained in the experiment. Some turbulence models even show a significant overestimation of the PL (e.g. RNG k–␧ model). This rather poor performance in simulating round turbulent jets is known as “round-jet anomaly” and is mainly due to the transport equation for the dissipation rate of turbulent kinetic energy (ε) [21,23]. For the standard k–ε model, Pope et al. [23] proposed a modified ε-equation including an additional term. The only model that showed a satisfying agreement with the experiment was the realizable k–ε (rke) model. A novel formulation of the ε-equation and an alternative approach to calculate the turbulent eddy viscosity (t ) are characterizing this rke model [21]. In what follows, the penetration lengths extracted from the mesh- and iteration-independent results of the numerical model (realizable k–ε turbulence model) are compared to the experimental values reported in [3]. In the first series of numerical simulations (referred to as series A), the mass flow rates of the co-flowing cooling water (65 g/s) and supercritical water (4 g/s) were kept conCW ) was set to 20 ◦ C. stant. The temperature of the cooling water (T∞ Hence, the influence of varying nozzle exit temperatures (T0 ) on the PL was investigated in this series. All conditions of the CFD simulations for series A are summarized in Table 2. Fig. 4(a) shows a comparison between experimental and numerical results (series A) in terms of the supercritical jet length (PL). A nozzle diameter of 3 mm was used for all data points. Simulations Table 2 Conditions applied in series A (see additionally Fig. 1). Nozzle exit temperature Temperature of cooling water SCW mass flow rate CW mass flow rate Nozzle exit Reynolds number Nozzle diameter Operating pressure

T0 CW T∞ ˙0 m ˙ CW m ∞ Re0 d0 pop

Varying: 376–982 20 4 65 ∼30,000 to ∼100,000 2.0, 3.0, 4.0 224

◦

C C g/s g/s – mm bar ◦

6 Experiment d0 = 3mm

5

Simulation d 0 = 3mm

4 3 2 1

(a)

Penetration length (PL) [mm]

Penetration length (PL) [mm]

M.J. Schuler et al. / J. of Supercritical Fluids 82 (2013) 213–220

0 360 380 400 420 440 460 480 500 520

5 Experiment d0 = 2mm

4

2 1 0 360 380 400 420 440 460 480 500 520

Nozzle exit temperature (T0) [°C] Fig. 5. Comparison between values of penetration length (PL) obtained by experiment and simulation for series A (see Table 2). A nozzle diameter of 2 mm and varying supercritical jet nozzle exit temperatures (T0 ) were used. The realizable k–ε model was applied.

Simulation d 0 = 3mm

4 3 2 1

(b)

0 500

600

700

800

900

1000

Nozzle exit temperature (T0) [°C] Fig. 4. (a) Comparison between values of penetration length (PL) obtained by experiment and simulation for series A (see Table 2). A nozzle diameter of 3 mm and varying supercritical jet nozzle exit temperatures (T0 ) were used. The realizable k–ε model was applied. (b) Simulated behavior of the PL with increasing nozzle exit temperatures exceeding the experimentally accessible range.

and experiments showed that the PL decreases with increasing nozzle exit temperature. Whereas this decrease in the PL is very pronounced in the vicinity of the PCT (375.21 ◦ C), the PL seems somehow to level off when moving toward higher temperatures. This effect is comprehensively described in [3] and could be predicted correctly by the numerical model. Also the absolute values of the PL extracted from the numerical flow simulations show a good agreement with the experiment. The weak local minimum of the PL found in the numerical values of Fig. 4(a) around 425 ◦ C and the afterwards slightly increasing trend in PL toward higher temperatures were not confirmed in the experiments. Due to the satisfactory agreement between experiment and simulation, the developed model was used to predict supercritical jet lengths beyond the experimentally accessible temperature range. These results include also two experimental data points at comparatively low temperatures as illustrated in Fig. 4(b). For the investigated nozzle exit temperatures (T0 ) far beyond the PCT, the PL was predicted to be virtually independent of T0 with a weak tendency of again decreasing values with increasing temperatures. The same comparison between simulation and experiment as made in Fig. 4 for a nozzle diameter of 3 mm is made in Fig. 5 for a nozzle diameter of 2 mm. In spite of a slight over-prediction of the PL by the simulations, the values and trends predicted by the simulation again agree satisfactorily with the experiments. There is, however, again a weak minimum in simulated PL around a temperature T0 of 425 ◦ C that was not confirmed in the measurements.

Also, the afterwards increasing PL with increasing nozzle exit temperatures was not evident in the experiments. Finally, the numerical model was also able to predict the experimental results for a nozzle diameter of 4 mm as seen in Fig. 6. The plot shows a good agreement between experiment and simulation with the calculated values for the PL being slightly below the experimental ones. Both, experiment and simulation show a consistent development of a decreasing PL with increasing nozzle exit temperatures for the investigated conditions. In the used rke turbulence model, the turbulent heat transfer is evaluated by the turbulent thermal conductivity (t = (cp t )/Prt ). The isobaric heat capacity (cp ), the turbulent viscosity (t ) and the turbulent Prandtl number (Prt ) are used to calculate this important quantity for predicting the cool down of SCW jets. Generally, due to the strong peak in the isobaric heat capacity respectively in the molecular Prandtl number (Pr = (cp )/), significantly enhanced turbulent heat transfer is predicted around the PCP of water. The general trends of series A observed in Figs. 4–6 are strongly related to the water’s thermo-physical properties. The significantly varying fluid properties around the PCT are responsible for the sharp decrease of the PL close to the PCP. According to Rothenfluh et al. [3], especially the density difference between the slowly

Penetration length (PL) [mm]

Penetration length (PL) [mm]

Experiment d0 = 3mm

5

Simulation d 0 = 2mm

3

Nozzle exit temperature (T0 ) [°C] 6

217

6 5 4 3 2 Experiment d0 = 4mm

1

Simulation d 0 = 4mm

0 360 380 400 420 440 460 480 500 520

Nozzle exit temperature (T0) [°C] Fig. 6. Comparison between values of penetration length (PL) obtained by experiment and simulation for series A (see Table 2). A nozzle diameter of 4 mm and varying supercritical jet nozzle exit temperatures (T0 ) were used. The realizable k–ε model was applied.

218

M.J. Schuler et al. / J. of Supercritical Fluids 82 (2013) 213–220

30

4 PCT (375.21°C)

8

PCT (375.21°C)

Series A

Series A

3 CW

[-]

20 15

8

H0/H

0

ρCW /ρ [-]

25

10

2

1 5 0 0

200

400

600

800

1000

Nozzle exit temperature (T0) [°C] Fig. 7. Density ratio between the slowly co-flowing cooling water and the hot jet water for varying nozzle exit temperatures (T0 ) at a pressure of 224 bar (see also Fig. 1). The calculated values are based on the conditions of series A (see Table 2). The REFPROP® (NIST) database was used to evaluate the thermo-physical properties.

co-flowing cooling water and the hot SCW at nozzle exit conditions is one of the main driving forces for the entrainment of cold CW / between water into the hot jet region. The density ratio ∞ 0 the cooling water (20 ◦ C, 224 bar) and the hot jet for varying nozzle exit temperatures (T0 , 224 bar) can be seen in Fig. 7. A sharp CW / can be observed around the increase of the density ratio ∞ 0 CW / in the vicinity of the PCP provokes a PCT. This behavior of ∞ 0 significant enhancement of entrainment and thus intensifies turbulent mixing between hot jet and surrounding cooling water with increasing nozzle exit temperatures. The fact that the PL around the PCP sharply drops in both, experiment and the simulation is a strong evidence for the improved heat transfer between hot jet and surrounding cooling water. For a fixed SCW mass flow rate (4 g/s for series A), the velocity at the nozzle exit position (u0 ) increases with increasing temperatures (T0 ) due to the density decrease. The momentum flow ratio between the SCW stream G0 = A0 · 0 · u0 2 and the cooling water 2

CW = ACW · CW · (uCW ) is exemplarily given in Fig. 8 for stream G∞ ∞ ∞ ∞ the investigated range of the nozzle diameter of 3 mm (see additionally Fig. 1). Apart from the density ratio illustrated in Fig. 7, it CW in Fig. 8 that contributes is also the observed increase of G0 /G∞ to enhanced entrainment and turbulent mixing at higher jet temperatures.

5 PCT (375.21°C) Series A, d0 = 3mm

3

8

G0 /G

CW

[-]

4

2

0 0

200

400

600

800

1000

Nozzle exit temperature (T0) [°C] ˙ 0 of the SCW jet normalized by the energy Fig. 9. Thermal energy flow H0 = h0 · m CW ˙ CW = hCW flow H∞ ∞ ·m ∞ of the co-flowing cooling water stream (see also Fig. 1). The calculated values are given in function of the nozzle exit temperature and are based on the conditions of series A (see Table 2). The REFPROP® (NIST) database was used to evaluate the thermo-physical properties.

In Fig. 9, the behavior of the ratio between the hot jet’s ther˙ 0 ) and the cooling water’s energy flow mal energy flow (H0 = h0 · m CW = hCW · m ˙ CW (H∞ ∞ ∞ ) with jet temperature (T0 ) is illustrated for the conditions of series A. Heating up the jet’s water and increasing thus the energy throughputs at the nozzle orifice, however, did not lead to the somehow expected significant elongation of jet’s penetration length as seen in Figs. 4–6 above. The reason for this behavior of the PL with the jet temperature (T0 ) can be found in the enhanced CW / and entrainment of cooling water owing to the increase of ∞ 0 CW as seen in Figs. 7 and 8. The enhanced entrainment and G0 /G∞ turbulent mixing at higher jet temperatures (T0 ) compensate for the increase of thermal energy injected through the nozzle. The additional thermal energy content of the jet with increasing nozzle exit temperatures is thus lost straightaway to the cold environment leading to generally slightly decreasing respectively rather constant values of the PL for the high nozzle exit temperatures (see e.g. Fig. 4(b)). In a second series B, the mass flow rate of the annular coolCW = 20 ◦ C) were again ing water (65 g/s) and its temperature (T∞ kept constant. Also the nozzle exit temperature of the SCW jet was adjusted to a constant value (T0 = 410 ◦ C). Therefore, unlike the simCW / was fixed to a constant ulations in series A, the density ratio ∞ 0 value of 8.74. In series B, the influence of the jet’s SCW mass flow ˙ 0 (varying quantity) on the PL was investigated. An overview rate m of the conditions applied in series B is given in Table 3. Figs. 10–12 show the results of series B for nozzle diameters of 3 mm (Fig. 10), 2 mm (Fig. 11) and 4 mm (Fig. 12). All plots show ˙ 0 ) comparing the PL as a function of the jet’s SCW mass flow rate (m the numerical results with the experimental ones. Generally, the performed numerical simulations were able to predict the experimental results to a satisfactory degree. The general trends in the experiments were also confirmed in the CFD simulations. Nevertheless, the slightly decreasing experimental PL with rising SCW

1 0 0

200

400

600

800

1000

Nozzle exit temperature (T0) [°C] CW Fig. 8. Momentum flow ratio G0 /G∞ between the SCW jet and the surrounding cooling water stream plotted versus the nozzle exit temperature (T0 ) (see also Fig. 1). The calculated values are based on the conditions of series A given in Table 2 for a nozzle diameter of 3 mm. The REFPROP® (NIST) database was used to evaluate the thermo-physical properties.

Table 3 Conditions applied in series B (see additionally Fig. 1). Nozzle exit temperature Temperature of cooling water SCW mass flow rate CW mass flow rate Nozzle exit Reynolds number Nozzle diameter Operating pressure

T0 CW T∞ ˙0 m ˙ CW m ∞ Re0 d0 pop

410 20 Varying: 1–10 65 ∼15,000 to ∼165,000 2.0, 3.0, 4.0 224

◦

C C g/s g/s – mm bar ◦

6

14

5

12

CW

[-]

4

8

G0 /G

3 2

Series B, d 0 = 3mm E 0 /E CW

8 6 PCT (375.21°C) G 0 /G CW 8

1

10

4

Experiment d0 = 3mm Simulation d 0 = 3mm

2 0

0 0

2

4

6

8

10

12

0

SCW mass flow rate (m0) [g/s]

Penetration length (PL) [mm]

5 Experiment d0 = 2mm

4

Simulation d 0 = 2mm

3 2 1 0 0

1

2

3

4

5

6

7

8

SCW mass flow rate (m0) [g/s] Fig. 11. Comparison between values of the penetration length (PL) obtained by experiment and simulation for series B (see Table 3). A nozzle diameter of 2 mm ˙ 0 ) were used. The realizable k–ε model was applied. and varying jet mass flows (m

mass flow rates could not be predicted by the simulations for the different nozzle diameters. ˙ 0 ) had a rather weak influence The jet’s SCW mass flow rate (m on the simulated PL within the experimentally accessible range of ˙ 0 , the thermal energy flow series B (see Figs. 10–12). By increasing m ˙ 0 ) through the nozzle orifice also increased linearly (H0 = h0 · m

6 5 4 3 2

Experiment d0 = 4mm Simulation d 0 = 4mm

1 0 0

1

2

3

4

5

2

4

6

8

10

12

14

SCW mass flow rate (m0) [g/s]

Fig. 10. Comparison between values of the penetration length (PL) obtained by experiment and simulation for series B (see Table 3). A nozzle diameter of 3 mm ˙ 0 ) were used. The realizable k–ε model was applied. and varying jet mass flows (m

Penetration length (PL) [mm]

219

8

Penetration length (PL) [mm]

M.J. Schuler et al. / J. of Supercritical Fluids 82 (2013) 213–220

6

SCW mass flow rate (m0) [g/s] Fig. 12. Comparison between values of the penetration length (PL) obtained by experiment and simulation for series B (see Table 3). A nozzle diameter of 4 mm ˙ 0 ) were used. The realizable k–ε model was applied. and varying jet mass flows (m

CW Fig. 13. Momentum flow ratio G0 /G∞ (see also Fig. 1) between the SCW jet and the surrounding cooling water stream plotted versus the SCW mass flow rate for the conditions of series B (see Table 3). A nozzle diameter of 3 mm was used. The REFPROP® (NIST) database was used to evaluate the thermo-physical properties of water.

˙ 0 . At the same time, however, the increasing momentum with m CW (see Fig. 13) leads to reinforced entrainment of flow ratio G0 /G∞ cooling water in the hot jet region. These two effects of enhanced entrainment and increase of the jet’s thermal energy seem to balance out each other within the experimental range of series B. This results in values of the PL that are nearly independent of the applied ˙ 0 ). SCW mass flow rates (m For high SCW mass flow rates out of the experimental range, the additional energy input seems to dominate the entrainment effects in series B. This finally resulted in a weak increase of the PL with ˙ 0 as illustrated in Figs. 10–12 for the different considered rising m nozzle diameters. Series A and series B presented above, both demonstrated that the introduced CFD model was able to satisfactorily predict the overall heat transfer due to entrainment effects from a submerged supercritical water jet to the cold surrounding water. 4. Conclusions A numerical model based on the commercial CFD tool ANSYS FLUENT® was developed and applied to round turbulent, supercritical water jets submerged in an environment of cool, subcritical water. After a converged solution had been obtained, the supercritical jet’s penetration length (PL) was extracted from the simulation and compared to experimental measurement. Generally, it could be shown that the model is able to predict the overall turbulent heat transfer between water in the gas-like state (supercritical water plume) and the surrounding cooling water in the liquid-like state (subcritical water) reasonably well. The mechanism mainly responsible for this heat transfer is the entrainment of cold surrounding cooling water into the central, hot jet region. The enhanced heat transfer around the pseudo-critical point of water due to the peak of the isobaric heat capacity additionally contribute to the cool down of the SCW jets. Apart from confirming the general experimental trends, the presented CFD model also predicted the absolute values of the supercritical penetration lengths (PL) found in the experiments with a reasonable accuracy. Generally, an increase in the jet’s nozzle exit temperature at a constant SCW mass flow rate (series A) leads to a decrease of the PL. This phenomenon can be explained with the entrainment of cooling water getting more intense at higher jet temperatures. The growing energy flows through the nozzle orifice for increasing nozzle exit temperatures cannot keep up with this enhanced cooling effect due to entrainment. Therefore, the supercritical jet length (PL) generally

220

M.J. Schuler et al. / J. of Supercritical Fluids 82 (2013) 213–220

decreases with increasing jet temperatures. At nozzle exit temperatures far beyond the PCT, however, the PL remains nearly unaffected by the jet’s temperature. A similar trend as described above could also be observed for simulations with a fixed nozzle exit temperature (410 ◦ C) and varying SCW mass flow rates (series B). Here, the PL of the jets remained nearly unchanged in the experimental range for the used nozzle ˙ 0 . Again, it was the entrainment being diameter with increasing m responsible for this result: The increasing energy flow through the nozzle at higher SCW mass flow rates is lost to the cold environment due the enhanced entrainment and turbulent mixing. These heat losses from the hot, supercritical water jet to the cold and dense environment have been detected as being one of the main challenges in hydrothermal spallation drilling. The possibility of predicting these crucial heat losses by the proposed CFD-model in the challenging numerical environment of strongly varying fluid properties is an important step in the development of the hydrothermal spallation drilling technology. The present work can contribute to optimize the design of a HSD drilling head by minimizing radial heat losses of the impinging supercritical water jets used for drilling. Acknowledgements This work was funded by Swisselectric Research and the Swiss National Science Foundation (SNF). We gratefully acknowledge this financial support. Moreover, the authors would like to thank P. Feusi, B. Kramer and M. Meuli for their technical support. References [1] W. Wagner, A. Pruss, The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use, Journal of Physical and Chemical Reference Data 31 (2) (2002) 387–535. [2] C.R. Augustine, Hydrothermal spallation drilling and advanced energy conversion technologies for engineered geothermal systems, Massachusetts Institute of Technology, 2009 (PhD thesis). [3] T. Rothenfluh, M.J. Schuler, P.R. von Rohr, Penetration length studies of supercritical water jets submerged in a subcritical water environment using a novel optical Schlieren method, The Journal of Supercritical Fluids 57 (2) (2011) 175–182. [4] M.D. Bermejo, M.J. Cocero, Supercritical water oxidation: a technical review, AIChE Journal 52 (11) (2006) 3933–3951. [5] K. Lieball, Numerical Investigations on a transpiring wall reactor for supercritical water oxidation, Institute of Process Engineering, No. 14’911, ETH Zurich, Zurich, Switzerland, 2003 (http://www.e-collection.ethz.ch). [6] C. Narayanan, et al., Numerical modelling of a supercritical water oxidation reactor containing a hydrothermal flame, Journal of Supercritical Fluids 46 (2) (2008) 149–155. [7] C. Augustine, J.W. Tester, Hydrothermal flames: from phenomenological experimental demonstrations to quantitative understanding, Journal of Supercritical Fluids 47 (3) (2009) 415–430.

[8] J. Bellan, Supercritical (and subcritical) fluid behavior and modeling: drops, streams, shear and mixing layers, jets and sprays, Progress in Energy and Combustion Science 26 (4–6) (2000) 329–366. [9] X. Cheng, T. Schulenberg, Heat transfer at supercritical pressures – Literature review and application to an HPLWR (FZKA 6609), Institut für Kern- und Energietechnik, Forschungszentrum Karlsruhe GmbH, Karlsruhe, 2001. [10] J.D. Jackson, W.B. Hall, Forced convection heat transfer to fluids at supercritical pressure, in: S. Kakac, D.B. Spalding (Eds.), Turbulent forced convection in channels and bundles, McGraw-Hill, New York, 1979. [11] J. Sierra-Pallares, P. Santiago-Casado, F. Castro, Numerical modelling of supercritical submerged water jets in a subcritical co-flow, The Journal of Supercritical Fluids 65 (0) (2012) 45–53. [12] M.J. Schuler, T. Rothenfluh, P. Rudolf von Rohr, Simulation of the thermal field of submerged supercritical water jets at near-critical pressures, The Journal of Supercritical Fluids 75 (0) (2013) 128–137. [13] P. Rudolf von Rohr, M. J. Schuler, and T. Rothenfluh., Rock drilling in great depths by thermal fragmentation using highly exothermic reactions evolving in the environment of a water-based drilling fluid., Patent pending, Patent number WO 2010/072407 A1. [14] F.W. Preston, H.E. White, Observations on spalling, Journal of the American Ceramic Society 17 (1934) 137–144. [15] R.M. Rauenzahn, J.W. Tester, Rock failure mechanisms of flame-jet thermal spallation drilling – theory and experimental testing, International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts 26 (5) (1989) 381–399. [16] R.E. Williams, The thermal spallation drilling process, Geothermics 15 (1) (1986) 17–22. [17] R.M. Rauenzahn, J.W. Tester, Numerical-simulation and field testing of flamejet thermal spallation drilling. 1. Model development, International Journal of Heat and Mass Transfer 34 (3) (1991) 795–808. [18] R.M. Rauenzahn, Analysis of Rock Mechanics and Gas Dynamics of Flame-Jet Thermal Spallation Drilling, Massachusetts Institute of Technology, 1986 (PhD thesis). [19] S.B. Pope, Turbulent Flows, Cambridge University Press, 2000. [20] H.K. Versteeg, W. Malalasekera, An Introduction to Computational Fluid Dynamics, 2nd ed., Pearson Education Limited, Edinburgh, 2007. [21] T.H. Shih, et al., A new kappa-epsilon eddy viscosity model for high Reynolds-number turbulent flows, Computers & Fluids 24 (3) (1995) 227–238. [22] E.W. Lemmon, M.L. Huber, M.O. McLinden, NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP, Version 8. 0, National Institute of Standards and Technology, Gaithersburg, 2007, Standard Reference Data Program. [23] S.B. Pope, Explanation of turbulent round-jet-plane-jet anomaly, AIAA Journal 16 (3) (1978) 279–281. [24] J. Sierra-Pallares, et al., Numerical modelling of hydrothermal flames. Micromixing effects over turbulent reaction rates, Journal of Supercritical Fluids 50 (2) (2009) 146–154. [25] J. Sierra-Pallares, et al., Numerical analysis of high-pressure fluid jets: application to RTD prediction in supercritical reactors, Journal of Supercritical Fluids 49 (2) (2009) 249–255. [26] K.W. Brinckman, W.H. Calhoon, S.M. Dash, Scalar fluctuation modeling for highspeed aeropropulsive flows, AIAA Journal 45 (5) (2007) 1036–1046. [27] R.E. Williams, R.M. Potter, S. Miska, Experiments in thermal spallation of various rocks, Journal of Energy Resources Technology-Transactions of the ASME 118 (1) (1996) 2–8. [28] M.A. Wilkinson, J.W. Tester, Experimental-measurement of surface temperatures during flame-jet induced thermal spallation, Rock Mechanics and Rock Engineering 26 (1) (1993) 29–62.